Good Budget Laptop For Writing

I'm looking for a budget laptop for writing. It doesn't need a great GPU or large hard drive, just something small lightweight and efficient for writing multiple hours at a time.

Size range is anywhere between 11 and 13 inches. My price range is anywhere between $300-500 AUD, and I'm more than happy to buy something second hand.

My main priorities are something with a fairly decent keyboard. I'm fine with a membrane board, as long as it feels ok. And of course, battery life, as I want to take it around with me.

All suggestions are welcome.